Rigid polyurethane foams are the better-performing material for the most common insulation purposes, like sandwich panels. Nevertheless, they are highly flammable materials, release toxic gases, and are manufactured from fossil sources. As an alternative, tannin foams are bio-based materials that work as innovative alternatives thanks to their great fire resistance, as well as lower smoke and harmful gases emissions. In the present study, lab-made foams of both materials were compared through morphology, thermal and fire degradation, mechanical properties, and water affinity in order to fill the technological gap between them and their related sandwich panels. It was observed that tannin foams are still relatively inhomogeneous (since formaldehyde was not used) and present a high affinity for water but have higher thermal and fire resistance. The flat compression strength of the polyurethane sandwiches was greater than that of tannin sandwiches (3.61 and 3.09 MPa, respectively) thanks, mainly, to the crosslinking degree difference between the resins. Also, tannin foams presented a lower weight loss (−70.684% lower weight loss in flammability tests than polyurethane foams) and the ability to self-extinguish the flame. Therefore, sandwich panels with tannin foam cores could be successful materials in areas that require protection against fire, such as the building engineering and automotive industries.